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Properties

Label 507.2.m.b.469.3
Level $507$
Weight $2$
Character 507.469
Analytic conductor $4.048$
Analytic rank $0$
Dimension $204$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [507,2,Mod(40,507)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("507.40"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(507, base_ring=CyclotomicField(26)) chi = DirichletCharacter(H, H._module([0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.m (of order \(13\), degree \(12\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [204] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.04841538248\)
Analytic rank: \(0\)
Dimension: \(204\)
Relative dimension: \(17\) over \(\Q(\zeta_{13})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{13}]$

Embedding invariants

Embedding label 469.3
Character \(\chi\) \(=\) 507.469
Dual form 507.2.m.b.40.3

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-2.29214 + 0.564962i) q^{2} +(0.120537 + 0.992709i) q^{3} +(3.16382 - 1.66050i) q^{4} +(-1.62559 - 2.35507i) q^{5} +(-0.837130 - 2.20733i) q^{6} +(3.18940 - 2.82556i) q^{7} +(-2.77972 + 2.46262i) q^{8} +(-0.970942 + 0.239316i) q^{9} +(5.05660 + 4.47975i) q^{10} +(1.08698 + 0.267916i) q^{11} +(2.02975 + 2.94060i) q^{12} +(-0.861346 - 3.50115i) q^{13} +(-5.71422 + 8.27847i) q^{14} +(2.14195 - 1.89761i) q^{15} +(0.920731 - 1.33391i) q^{16} +(-1.95440 + 1.73145i) q^{17} +(2.09033 - 1.09709i) q^{18} -7.80578 q^{19} +(-9.05365 - 4.75172i) q^{20} +(3.18940 + 2.82556i) q^{21} -2.64287 q^{22} -5.59591 q^{23} +(-2.77972 - 2.46262i) q^{24} +(-1.13079 + 2.98165i) q^{25} +(3.95235 + 7.53851i) q^{26} +(-0.354605 - 0.935016i) q^{27} +(5.39883 - 14.2356i) q^{28} +(-0.848092 + 0.209036i) q^{29} +(-3.83759 + 5.55970i) q^{30} +(0.167191 + 0.440847i) q^{31} +(1.27693 - 3.36700i) q^{32} +(-0.134942 + 1.11135i) q^{33} +(3.50156 - 5.07288i) q^{34} +(-11.8390 - 2.91806i) q^{35} +(-2.67450 + 2.36940i) q^{36} +(2.71345 + 7.15477i) q^{37} +(17.8920 - 4.40997i) q^{38} +(3.37180 - 1.27708i) q^{39} +(10.3183 + 2.54323i) q^{40} +(-1.00198 - 8.25206i) q^{41} +(-8.90688 - 4.67469i) q^{42} +(-2.95742 + 7.79807i) q^{43} +(3.88388 - 0.957290i) q^{44} +(2.14195 + 1.89761i) q^{45} +(12.8266 - 3.16148i) q^{46} +(4.90232 + 2.57294i) q^{47} +(1.43516 + 0.753233i) q^{48} +(1.34471 - 11.0747i) q^{49} +(0.907412 - 7.47321i) q^{50} +(-1.95440 - 1.73145i) q^{51} +(-8.53881 - 9.64675i) q^{52} +(4.53600 - 4.01855i) q^{53} +(1.34105 + 1.94285i) q^{54} +(-1.13602 - 2.99543i) q^{55} +(-1.90736 + 15.7085i) q^{56} +(-0.940883 - 7.74887i) q^{57} +(1.82585 - 0.958280i) q^{58} +(-5.96472 - 8.64139i) q^{59} +(3.62578 - 9.56039i) q^{60} +(-9.62202 - 8.52437i) q^{61} +(-0.632288 - 0.916028i) q^{62} +(-2.42052 + 3.50673i) q^{63} +(-1.41542 + 11.6571i) q^{64} +(-6.84526 + 7.71995i) q^{65} +(-0.318563 - 2.62360i) q^{66} +(-7.90439 - 4.14854i) q^{67} +(-3.30830 + 8.72326i) q^{68} +(-0.674512 - 5.55511i) q^{69} +28.7853 q^{70} +(0.0439480 + 0.361944i) q^{71} +(2.10961 - 3.05629i) q^{72} +(8.01763 + 1.97617i) q^{73} +(-10.2618 - 14.8668i) q^{74} +(-3.09621 - 0.763147i) q^{75} +(-24.6961 + 12.9615i) q^{76} +(4.22382 - 2.21683i) q^{77} +(-7.00715 + 4.83220i) q^{78} +(-3.54228 - 1.85913i) q^{79} -4.63817 q^{80} +(0.885456 - 0.464723i) q^{81} +(6.95879 + 18.3488i) q^{82} +(-0.647852 + 5.33554i) q^{83} +(14.7825 + 3.64356i) q^{84} +(7.25472 + 1.78813i) q^{85} +(2.37321 - 19.5451i) q^{86} +(-0.309738 - 0.816712i) q^{87} +(-3.68127 + 1.93208i) q^{88} -2.41731 q^{89} +(-5.98174 - 3.13946i) q^{90} +(-12.6399 - 8.73279i) q^{91} +(-17.7044 + 9.29201i) q^{92} +(-0.417480 + 0.219111i) q^{93} +(-12.6904 - 3.12791i) q^{94} +(12.6890 + 18.3831i) q^{95} +(3.49637 + 0.861777i) q^{96} +(7.46799 - 10.8193i) q^{97} +(3.17451 + 26.1445i) q^{98} -1.11951 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 204 q - q^{2} - 17 q^{3} - 21 q^{4} - 6 q^{5} - q^{6} - 8 q^{7} - 9 q^{8} - 17 q^{9} - 6 q^{10} - 8 q^{11} - 21 q^{12} + 54 q^{13} - 30 q^{14} - 6 q^{15} - 45 q^{16} - 18 q^{17} - q^{18} - 20 q^{19}+ \cdots + 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/507\mathbb{Z}\right)^\times\).

\(n\) \(170\) \(340\)
\(\chi(n)\) \(1\) \(e\left(\frac{12}{13}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.29214 + 0.564962i −1.62079 + 0.399489i −0.942206 − 0.335035i \(-0.891252\pi\)
−0.678583 + 0.734524i \(0.737406\pi\)
\(3\) 0.120537 + 0.992709i 0.0695919 + 0.573141i
\(4\) 3.16382 − 1.66050i 1.58191 − 0.830250i
\(5\) −1.62559 − 2.35507i −0.726984 − 1.05322i −0.995936 − 0.0900646i \(-0.971293\pi\)
0.268952 − 0.963154i \(-0.413323\pi\)
\(6\) −0.837130 − 2.20733i −0.341757 − 0.901139i
\(7\) 3.18940 − 2.82556i 1.20548 − 1.06796i 0.209552 − 0.977798i \(-0.432800\pi\)
0.995927 − 0.0901634i \(-0.0287389\pi\)
\(8\) −2.77972 + 2.46262i −0.982780 + 0.870667i
\(9\) −0.970942 + 0.239316i −0.323647 + 0.0797719i
\(10\) 5.05660 + 4.47975i 1.59904 + 1.41662i
\(11\) 1.08698 + 0.267916i 0.327736 + 0.0807797i 0.399751 − 0.916624i \(-0.369097\pi\)
−0.0720146 + 0.997404i \(0.522943\pi\)
\(12\) 2.02975 + 2.94060i 0.585938 + 0.848878i
\(13\) −0.861346 − 3.50115i −0.238894 − 0.971046i
\(14\) −5.71422 + 8.27847i −1.52719 + 2.21251i
\(15\) 2.14195 − 1.89761i 0.553050 − 0.489960i
\(16\) 0.920731 − 1.33391i 0.230183 − 0.333477i
\(17\) −1.95440 + 1.73145i −0.474011 + 0.419937i −0.866009 − 0.500029i \(-0.833323\pi\)
0.391997 + 0.919966i \(0.371784\pi\)
\(18\) 2.09033 − 1.09709i 0.492696 − 0.258587i
\(19\) −7.80578 −1.79077 −0.895385 − 0.445294i \(-0.853099\pi\)
−0.895385 + 0.445294i \(0.853099\pi\)
\(20\) −9.05365 − 4.75172i −2.02446 − 1.06252i
\(21\) 3.18940 + 2.82556i 0.695983 + 0.616588i
\(22\) −2.64287 −0.563462
\(23\) −5.59591 −1.16683 −0.583414 − 0.812175i \(-0.698283\pi\)
−0.583414 + 0.812175i \(0.698283\pi\)
\(24\) −2.77972 − 2.46262i −0.567408 − 0.502680i
\(25\) −1.13079 + 2.98165i −0.226158 + 0.596330i
\(26\) 3.95235 + 7.53851i 0.775119 + 1.47842i
\(27\) −0.354605 − 0.935016i −0.0682437 − 0.179944i
\(28\) 5.39883 − 14.2356i 1.02028 − 2.69027i
\(29\) −0.848092 + 0.209036i −0.157487 + 0.0388170i −0.317271 − 0.948335i \(-0.602766\pi\)
0.159784 + 0.987152i \(0.448920\pi\)
\(30\) −3.83759 + 5.55970i −0.700644 + 1.01506i
\(31\) 0.167191 + 0.440847i 0.0300284 + 0.0791785i 0.949192 − 0.314698i \(-0.101903\pi\)
−0.919164 + 0.393876i \(0.871134\pi\)
\(32\) 1.27693 − 3.36700i 0.225732 − 0.595207i
\(33\) −0.134942 + 1.11135i −0.0234904 + 0.193461i
\(34\) 3.50156 − 5.07288i 0.600512 − 0.869992i
\(35\) −11.8390 − 2.91806i −2.00116 − 0.493242i
\(36\) −2.67450 + 2.36940i −0.445750 + 0.394900i
\(37\) 2.71345 + 7.15477i 0.446088 + 1.17624i 0.949936 + 0.312445i \(0.101148\pi\)
−0.503848 + 0.863793i \(0.668083\pi\)
\(38\) 17.8920 − 4.40997i 2.90246 − 0.715392i
\(39\) 3.37180 − 1.27708i 0.539921 − 0.204497i
\(40\) 10.3183 + 2.54323i 1.63147 + 0.402121i
\(41\) −1.00198 − 8.25206i −0.156483 − 1.28876i −0.833389 − 0.552686i \(-0.813603\pi\)
0.676906 − 0.736069i \(-0.263320\pi\)
\(42\) −8.90688 − 4.67469i −1.37436 − 0.721321i
\(43\) −2.95742 + 7.79807i −0.451002 + 1.18919i 0.496101 + 0.868265i \(0.334765\pi\)
−0.947103 + 0.320930i \(0.896005\pi\)
\(44\) 3.88388 − 0.957290i 0.585516 − 0.144317i
\(45\) 2.14195 + 1.89761i 0.319304 + 0.282878i
\(46\) 12.8266 − 3.16148i 1.89118 − 0.466134i
\(47\) 4.90232 + 2.57294i 0.715077 + 0.375302i 0.782691 − 0.622410i \(-0.213846\pi\)
−0.0676140 + 0.997712i \(0.521539\pi\)
\(48\) 1.43516 + 0.753233i 0.207148 + 0.108720i
\(49\) 1.34471 − 11.0747i 0.192102 − 1.58210i
\(50\) 0.907412 − 7.47321i 0.128327 − 1.05687i
\(51\) −1.95440 − 1.73145i −0.273671 − 0.242451i
\(52\) −8.53881 − 9.64675i −1.18412 − 1.33776i
\(53\) 4.53600 − 4.01855i 0.623068 − 0.551990i −0.291424 − 0.956594i \(-0.594129\pi\)
0.914492 + 0.404604i \(0.132591\pi\)
\(54\) 1.34105 + 1.94285i 0.182494 + 0.264389i
\(55\) −1.13602 − 2.99543i −0.153180 − 0.403903i
\(56\) −1.90736 + 15.7085i −0.254882 + 2.09914i
\(57\) −0.940883 − 7.74887i −0.124623 − 1.02636i
\(58\) 1.82585 − 0.958280i 0.239746 − 0.125828i
\(59\) −5.96472 − 8.64139i −0.776541 − 1.12501i −0.989010 − 0.147846i \(-0.952766\pi\)
0.212470 − 0.977168i \(-0.431849\pi\)
\(60\) 3.62578 − 9.56039i 0.468086 − 1.23424i
\(61\) −9.62202 − 8.52437i −1.23197 − 1.09143i −0.992716 − 0.120477i \(-0.961558\pi\)
−0.239257 − 0.970956i \(-0.576904\pi\)
\(62\) −0.632288 − 0.916028i −0.0803007 − 0.116336i
\(63\) −2.42052 + 3.50673i −0.304957 + 0.441806i
\(64\) −1.41542 + 11.6571i −0.176928 + 1.45713i
\(65\) −6.84526 + 7.71995i −0.849051 + 0.957543i
\(66\) −0.318563 − 2.62360i −0.0392124 − 0.322943i
\(67\) −7.90439 − 4.14854i −0.965675 − 0.506825i −0.0932867 − 0.995639i \(-0.529737\pi\)
−0.872388 + 0.488814i \(0.837430\pi\)
\(68\) −3.30830 + 8.72326i −0.401190 + 1.05785i
\(69\) −0.674512 − 5.55511i −0.0812017 − 0.668757i
\(70\) 28.7853 3.44050
\(71\) 0.0439480 + 0.361944i 0.00521567 + 0.0429549i 0.995084 − 0.0990310i \(-0.0315743\pi\)
−0.989869 + 0.141986i \(0.954651\pi\)
\(72\) 2.10961 − 3.05629i 0.248619 − 0.360187i
\(73\) 8.01763 + 1.97617i 0.938393 + 0.231293i 0.678715 − 0.734402i \(-0.262537\pi\)
0.259678 + 0.965695i \(0.416383\pi\)
\(74\) −10.2618 − 14.8668i −1.19291 − 1.72823i
\(75\) −3.09621 − 0.763147i −0.357519 − 0.0881206i
\(76\) −24.6961 + 12.9615i −2.83283 + 1.48679i
\(77\) 4.22382 − 2.21683i 0.481349 − 0.252631i
\(78\) −7.00715 + 4.83220i −0.793403 + 0.547139i
\(79\) −3.54228 − 1.85913i −0.398538 − 0.209169i 0.253533 − 0.967327i \(-0.418407\pi\)
−0.652071 + 0.758158i \(0.726100\pi\)
\(80\) −4.63817 −0.518563
\(81\) 0.885456 − 0.464723i 0.0983840 − 0.0516359i
\(82\) 6.95879 + 18.3488i 0.768469 + 2.02629i
\(83\) −0.647852 + 5.33554i −0.0711110 + 0.585652i 0.912982 + 0.407999i \(0.133773\pi\)
−0.984093 + 0.177653i \(0.943150\pi\)
\(84\) 14.7825 + 3.64356i 1.61290 + 0.397545i
\(85\) 7.25472 + 1.78813i 0.786884 + 0.193950i
\(86\) 2.37321 − 19.5451i 0.255909 − 2.10760i
\(87\) −0.309738 − 0.816712i −0.0332074 − 0.0875607i
\(88\) −3.68127 + 1.93208i −0.392425 + 0.205960i
\(89\) −2.41731 −0.256235 −0.128117 − 0.991759i \(-0.540893\pi\)
−0.128117 + 0.991759i \(0.540893\pi\)
\(90\) −5.98174 − 3.13946i −0.630530 − 0.330928i
\(91\) −12.6399 − 8.73279i −1.32502 − 0.915445i
\(92\) −17.7044 + 9.29201i −1.84582 + 0.968759i
\(93\) −0.417480 + 0.219111i −0.0432907 + 0.0227207i
\(94\) −12.6904 − 3.12791i −1.30892 − 0.322619i
\(95\) 12.6890 + 18.3831i 1.30186 + 1.88607i
\(96\) 3.49637 + 0.861777i 0.356846 + 0.0879547i
\(97\) 7.46799 − 10.8193i 0.758260 − 1.09853i −0.233741 − 0.972299i \(-0.575097\pi\)
0.992001 − 0.126230i \(-0.0402877\pi\)
\(98\) 3.17451 + 26.1445i 0.320674 + 2.64099i
\(99\) −1.11951 −0.112515
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 507.2.m.b.469.3 yes 204
169.40 even 13 inner 507.2.m.b.40.3 ✓ 204
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
507.2.m.b.40.3 ✓ 204 169.40 even 13 inner
507.2.m.b.469.3 yes 204 1.1 even 1 trivial